- Email: [email protected]
Fatigue crack growth simulation under cyclic non-proportional mixed mode loading
Accepted Manuscript Fatigue crack growth simulation under cyclic non-proportional mixed mode loading Ying Yang, Michael Vormwald PII: DOI: Reference:
S0142-1123(17)30195-0 http://dx.doi.org/10.1016/j.ijfatigue.2017.04.014 JIJF 4324
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
21 February 2017 26 April 2017 27 April 2017
Please cite this article as: Yang, Y., Vormwald, M., Fatigue crack growth simulation under cyclic non-proportional mixed mode loading, International Journal of Fatigue (2017), doi: http://dx.doi.org/10.1016/j.ijfatigue.2017.04.014
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Fatigue crack growth simulation under cyclic non-proportional mixed mode loading Ying Yang a,b, c*, Michael Vormwald b a
School of Automotive Engineering, Wuhan University of Technology, Wuhan, Hubei, 430070, China b Materials Mechanics Group, Technische Universität Darmstadt, Franziska-Braun-Str. 3, D-64287 Darmstadt, Germany c Hubei Key Laboratory of Advanced Technology for Automotive Components, Wuhan, Hubei, 430070, China Abstract An algorithm based on linear elastic fracture mechanics for three-dimensional fatigue crack growth simulation under non-proportional mixed-mode loading is proposed in the present paper. The crack growth behaviour in thin-walled, hollow cylinders with a notch under combined non-proportional cyclic tension and torsion loadings are investigated with the finite element program ABAQUS and a 3D fracture analysis software FRANC3D. Different mixed-mode crack path prediction criteria are evaluated to estimate the crack growth direction. Fracture mode transition is observed, some influence factors are discussed. Crack growth cycles calculated based on the effective stress intensity factors satisfactorily match the experimental data for specimens tested under lower loading levels. Keywords Fatigue crack growth, Linear elastic fracture mechanics, Non-proportional loading, Mixed mode
*Corresponding author: Ying Yang Tel: +86 18670810648 E-mail address: [email protected]
1. Introduction When a crack encounters a mixed mode loading, the crack growth path differs from the path under a normal tension loading case, and a deviation will occur. Research in this topic has encountered many problems and challenges in both the theoretical and the experimental fields. In recent years, a series of criteria have been proposed for predictions of the crack propagation direction under mixed mode loading. This paper attempts to investigate and explain fatigue crack growth behavior under non-proportional mixed mode loadings, including the crack growth path and fatigue life assessment. Since Erdogan and Sih [3] first proposed the maximum tangential stress (MTS) criterion for mixed mode fracture prediction in the year of 1963, research in the field of mixed mode crack growth has attracted more and more attention. The minimum strain energy density criterion [4] (S-Criterion, Sih 1974), the maximum energy release rate criterion [5] (Nuismer, 1975), the maximum tangential strain criterion [6] (Chambers, 1991) and so on were suggested subsequently. The various criteria do not provide largely different results for the predicted crack propagation paths under monotonic mixed mode loading. This conclusion is supported by a large quantity of experimental data and theoretical analyses [7] [8] (Maiti, 1983; Maiti, 1984), if the cracks growth is controlled by mode I dominated (tension mode) fracture. However, large distinction between crack propagation direction and the predicted crack kink angle were also observed in some recent experiments. In these cases, the crack growth occurred in the plane where KII values are maximum. As a result the maximum shear stress (MSS) criterion [9] was proposed subsequently (Maccagno, 1992). MSS criterion is available for the predictions of crack growth driven by mode II dominated fracture (shear mode). Although all the above mentioned mixed mode fracture criteria have been deduced from monotonic loading case, they can be implemented straightforward to cyclic proportional mixed mode loading [10] (Highsmith, 2009). According to Bold et al. [11] (1992), this is due to the maximum mechanical parameter ranges (stress or strain) under proportional loading is in direct proportion to the maximum value of these parameters in the crack tip region. Fatigue crack growth under non-proportional mixed mode loading is a more complex situation and there are insufficient hypotheses for application. The very limited researches in this field generally focus on experimental analysis. Seven relevant factors which influence the crack growth behavior under non-proportional loading have been identified in a recent review by Vormwald and Zerres [12] (2012). In the present paper, some of these factors (such as mode-mixity, crack closure and different phase angle) are taken into account by an approach based on linear elastic fracture mechanics.
2. Experiment and material information The current paper focuses on the numerical simulation of 3-dimensional fatigue crack growth under non-proportional mixed mode loading. The corresponding
experimental data were obtained by Brüning [13] (2008). They have been discussed by Zerres et. al [14] [15] (2010, 2011). The aluminium alloy AlMg4.5Mn specimens and steel S460N specimens were tested with the geometry as shown in figure 1. The material properties are shown in the following Table 1[13] [16] [21] (Brüning, 2008; Vormwald,1986; Vormwald,2011). Combined non-proportional cyclic tension and torsion loading with phase angles of 45° and 90° were applied to the specimens with an R ratio of -1. The results of 2 aluminium specimens marked as A7 and A8 and 2 steel specimens marked as S13 and S7 are discussed. 3. Simulation procedure Figure 2 illustrates the procedure of this algorithm. Three main modules can be considered as: (a) determining the crack initiation position; (b) calculating of the maximum equivalent stress intensity factor Keq in one load cycle, which is taken as the crack driving force parameter; (c) crack growth process. Module (b) and (c) are repeated until the crack growth paths can be presented clearly. 3.1 Location of crack The crack initiation will occur in the notch root at the position where the locally uniaxial notch stress is maximum in one loading cycle. A 3-dimensional finite element model was created using ABAQUS for stress analysis. The mesh was refined in the crack growth region as shown in Fig. 3 (a). Along the notch root, there are four potential crack initiation sites numbered as 1, 2, 3 and 4 clockwise, shown in Fig.3 (b). The analysis indicated that the maximum tangential stress in one loading cycle along the notch root on the inside edges is almost 2 times larger than the tangential stress on the outside edge. Therefore a corner crack is considered to be a reasonable shape to initial crack simulation. As can be expected, the initial corner crack will propagate through the thickness of the notch and extend to the outside surface of the specimen in the crack growth process. An incipient corner crack was inserted into the inside edge of the notch root where the tangential stress is maximum by program Fracture Analysis Code 3D (FRANC3D), see figure 4. This program package is designed to simulate crack growth in engineering structures or components with arbitrary crack geometry under different loading and boundary conditions. FRANC3D adaptively remeshes an existing finite element model with the inserted initial crack to generate a cracked model. Furthermore, FRANC3D is also used to calculate stress intensity factor (SIF) for an entire crack front. 3.2 Crack driving force calculation The uncracked finite element model in the current simulation was created in ABAQUS. The typical working flow for ABAQUS/FRANC3D analysis is illustrated in figure 5. In ABAQUS the whole model is divided as the local model and the global model. Boundary conditions and loadings are applied on the global model. The initial cracks are inserted in the local model and cracks only grow in the local model,
so the local model is also called as the crack growth region. The local model is inputted to the FRANC3D to insert the initial cracks in the first step and to calculate SIFs, extend the cracks in the following steps.The cracked local model needs to be merged with the global model, so that the stress analysis can be performed in ABAQUS. Stress analysis results for the cracked model from ABAQUS are used to calculate stress intensity factors in FRANC3D. In the present paper, mode I and mode II SIFs, KI and KII in one cycle under non-proportional tension combined with torsion loading are computed for the sake of obtaining the equivalent SIF, Keq. Erdogan and Sih proposed that the equivalent stress intensity factor according to the maximum tangential stress criterion can be described by:
1 * 3 * 3 * 3 * Keq 3cos cos K sin sin K 4 2 2 4 2 2
(1)
θ* is a function of KI and KII, representing the direction where the tangential stress σθθ in the vicinity of the crack tip is maximum. And also the shear stress τrθ vanishes along this direction on the basis of the MTS criterion. 2 1 K 1 K 2arctan 8 4 K 4 K *
(2)
The stress intensity factors for mode I and mode II are calculated by FRANC3D software for a unit load in tension F=1 and in torsion MT=1. The combined tension and torsion mixed mode stress intensity factors KI and KII are obtained from the superposition according to equation (3):
K I t K I,F F t K I,M M T t K II t K II,F F t K II,M M T t
(3)
The combined SIFs computed from equation (3) are inserted to equation (2) to find the angle θ*, which is put into equation (1). The equivalent stress intensity factor Keq is calculated for a full loading cycle. Among all the equivalent stress intensity factors, the peak value Keq,max is taken as the crack driving force, which means the loads at this instant tmax, F(tmax) and MT(tmax) are used for crack propagation simulation. 3.3 Crack growth process The load corresponding to the Keq,max, F(tmax) and MT(tmax) are applied to the cracked finite element model. Stress analysis is carried out in ABAQUS. The SIF solutions for the cracks are again obtained from FRANC3D based on the stress analysis. Although the solutions for KI, KII and Keq are already at hand for t= tmax, the
re-evaluation of this loading case combination can now be used to let the FRANC3D software create the crack propagation increment. In FRANC3D, a 3-dimensional crack front is a space curve comprising a series of extrapolated nodes. For each node, the corresponding increment Δanode i is a value in a given proportion to a user defined crack growth increment Δauser. This proportion is related to the stress intensity factors along the crack front. A new node is extended in front of the old node with a distance of Δanode i as shown in figure 6 calculated based on equation (4).
Δanode i
ΔKnode i = Δauser ΔKeq,mean
n
(4)
The ABAQUS/FRANC3D interface was implemented to perform the crack growth analysis. The finite element modeling and the stress analysis are performed in ABAQUS, FRANC3D is used to calculate crack growth parameters and updates the crack geometry and mesh. This process is continued until the crack has grown to a certain length. The KI and KII of a node in the crack front and the KI and KII for the whole crack front under F(tmax) and MT(tmax) for the initial crack of specimen A8 are shown in figure 7. For the intial crack of specimen A8, the maximum mode I stress intensity factor component is more than 10 times larger than the maximum mode II values. This indicates that the mode II fracture will not affect crack growth behavior in the full cycle, mode I fracture is determined. For this reason, the MTS criterion is used as the primary rule for estimating the crack propagation direction in the crack growth simulation. The MSS criterion is performed as an alternative formula applicable to mode II controlled crack growth. In some cases, the simulated crack growth path deviated from the experimental crack path no matter what criterion was employed. In order to investigate the crack growth path behavior under these complicated loading cases, a modification is designed to keep the deviation between experimental results and simulation in an acceptable extent. The modification is implemented manually to adjust the position of the newly generated crack front in order to make the simulated crack increment direction coincident with the experimental results. As a result, although the simulated crack growth path is not perfectly identical with the experimental data, a close match is enforced. 4. Crack growth paths 4.1 Non-proportional loading with phase angle of 45° An aluminum specimen labeled A8 (AlMg4.5Mn; Fmax = 6.5 kN; MTmax = 72 Nm) was tested under non-proportional mixed mode loading with a phase angle of 45°. Figure 8(a) shows the crack propagation paths on the outer surface of the specimen in the experiment and in the simulation. As mentioned before, four potential crack initiation sites at the notch root were
numbered 1, 2, 3, and 4 clockwise (see figure 3). The corresponding cracks are called crack 1, 2, 3, and 4, respectively. Two crack paths were observed in specimen A8. Crack 1 is shown on the left of figure 8(a), and crack 3 is on the right. Cracks initiated at sites 1 and 3 because the maximum tangential stresses at these two sites were larger than those at sites 2 and 4 in one loading cycle. In the simulation, the initiation angle was approximately 38.4° (-38.4°). As can be seen in the figure, the crack propagation trajectories in the simulation were similar to those in the experiment. Cracks propagated along an approximate angle of 40° to the longitudinal axis of the specimen initially and then deflected gradually until the paths transformed into planes, which were perpendicular to the longitudinal axis. Because the two crack growth trajectories showed antisymmetry in both the experiment and the simulation, the behaviors of these two cracks should coincide. Taking crack 3 to analyze the crack path behavior, the simulated crack 3 can be seen in figure 9. Three stages of the crack path are shown in this figure: The crack path from notch root to point A was predicted by the MTS criterion in FRANC3D; that from point B to the end was predicted by the MSS criterion in FRANC3D; and the path AB was fitted from the experimental result. To provide a clear comparison between the paths predicted by the different criteria, two estimated results are also shown in figure 9. The path indicated by squares is the estimated result calculated based on the MTS criterion, and the path indicated by triangles is that based on the MSS criterion. Until point A, the crack propagation trajectory was satisfactorily estimated by the MTS criterion. When the MTS criterion was applied continuously, the estimated solution showed a downward crack path propagating in nearly the original direction with slight deflection. However, from point A, the real crack path deviated from the prediction based on the MTS criterion. This implies that the crack propagation direction was no longer normal to the maximum tangential stress in the vicinity of the crack tip. Another criterion was therefore used to attempt to calculate the crack kink angle from this point. The MSS criterion was applied here as an alternative rule to predict the crack path. The crack path starting from point A clearly did not follow the route estimated by either the MTS criterion or the MSS criterion based on the estimated solutions. The path according to the MSS criterion showed an upward trend from the original crack growth direction. A path somewhere between the predictions of the MTS and MSS criteria that gradually deviated from the original direction appears to be more reasonable. At present, no criteria are available that can be used to calculate crack path AB. As a result, path AB in FRANC3D is a fitting curve based on the experimental data. From point B, the crack path flattened (perpendicular to the longitudinal axis of the specimen), and the MSS criterion provided a successful prediction. The crack path calculated in FRANC3D matched the real crack path well. In contrast, large divergence was observed when the MTS criterion was used to predict the crack path, as shown in figure 9, from the estimated solution. Generally, the MTS criterion predicted that the crack path maintained a nearly straight line with a constant angle to the longitudinal axis, with slight deflection. The flat crack path was satisfactorily described by the MSS criterion. As we already
know, based on the LEFM analysis, the MTS criterion is valid for tensile mode fractures, and the MSS criterion is used for shear mode problems. Both of these criteria are supported by a large quantity of corresponding experiments. Based on the crack path behavior analysis, two distinct fracture modes and fracture mode transition were observed in the experiment. A steel specimen labeled S13 (S460N; Fmax = 22.5 kN; MTmax = 272 Nm) was subjected to cyclic tension combined with torsion loading with an out-of-phase angle of 45°. The crack paths in the experiment and those in the simulation are shown in figure 8(b). Crack 1 is on the left of this figure, and crack 3 is on the right. The crack propagation trajectories were similar to the paths in specimen A8. Cracks initiated at sites 1 and 3. The angle ϕ calculated based on finite element stress analysis was approximately 40.6° (-40.6°). Initially, both cracks propagated along an angle of approximately 40° (-40°) to the longitudinal axis. The final paths flattened and were perpendicular to the longitudinal axis of the specimen. Fracture mode transition from tensile to shear was also observed in this specimen. 4.2 Non-proportional loading with phase angle of 90° The other non-proportional loading tested in the experiment was with an out-of-phase loading angle of 90°. Figure 10(a) shows photographs of crack paths in an aluminum specimen labeled A7 (AlMg4.5Mn; Fmax = 8.0 kN; MTmax = 96 Nm) and the simulated crack paths. Two cracks were observed: crack 1 (on the left in the figure) and crack 3 (on the right of the figure). From the photographs, it was found that both crack paths were very curvilinear, and crack branching was observed in the experiment. When this test was performed, the simulation algorithm did not cover crack branching, and therefore, the simulation stopped before any crack branching could occur. According to the stress analysis, the crack initiation angle ϕ was 51.9°. Taking crack 1 to analyze the crack path behavior, the path first propagated upward following a smooth curve, and as the crack grew, the extension direction changed to downward, which was opposite to the original orientation. The crack paths in this specimen were different from those in specimens under loading with a phase angle of 45°. No flat crack path was observed in the experiment. The result of analysis of crack path behavior is shown in figure 11. From the notch root to point A, the smooth crack path can be described well by the MTS criterion. The corresponding fracture was tensile mode fracture. Deviations between the simulated path and the experimental path became evident as the crack extended. Starting from point A, the MSS criterion was used as the alternative rule to predict crack growth direction. Estimation shows that both the prediction from the MTS criterion and that from the MSS criterion did not provide acceptable results, because the real crack path was somewhere between these two estimations. When the crack length reached point B, the downward crack path was fully controlled by the MSS criterion. From this moment, the crack was controlled under shear mode fracture. Path AB was considered to signify transition mode and was fitted from the
experimental data. Steel specimen S7 was tested under tension combined with torsion loading with Fmax = 27 kN and MTmax = 408 Nm. Four cracks were observed in the experiment. In specimen S7, four crack paths were observed to be nearly symmetrical in both the experiment and the simulation, as shown in figure 10(b). According to the FEA, the maximum tangential stresses along the notch root at sites 1to 4 were nearly equal. The calculated crack initiation angle was 53.1° (-53.1°). Because the four cracks initiated at nearly the same loading cycles in the experiment, initial cracks of the same size were inserted into the corresponding positions in FRANC3D. The entire simulated crack path was calculated from FRANC3D based on the MTS criterion, which showed that crack propagation complied with tension controlled fracture, and no fracture mode transition took place. 5. Crack growth cycles Fatigue life includes the loading cycles expended in crack initiation and those expended during crack propagation. In the present paper, the number of cycles expended from the initial crack to that of the final crack is calculated based on Paris law, the number of loading cycles consumed in crack initiation is obtained from experimental data. 5.1 Crack closure effect The maximum equivalent stress intensity factor Keq,max in one loading cycle is used to calculate the instant forces corresponding to crack propagation. Under cyclic loading, the stress intensity factor range ΔK is associated with the crack deflection angle as well as fatigue life. When loading ratio R = -1, crack closure must affect fatigue life significantly. Usually, ΔK is substituted for the effective stress intensity factor range ΔKeff to take account of the loading ratio effect. Here, ΔKeff is defined as Kmax − Kop, where the subscript “op” denotes a crack opening stress intensity factor. Newman’s crack opening stress equation [18] is one of the most widely used methods for estimating opening stress and is given by: σop σmax σop σmax
= A0 + A1 R + A2 R2 + A3 R3
for R > 0
(5) for - 2 R < 0
= A0 + A1 R
Where
πσ A0 = 0.825 - 0.34α + 0.05α2 cos max 2σ0 A1 = (0.415 - 0.071α)
A2 = 1- A0 - A1 - A3
σmax σ0
1
α
(6)
(7)
(8)
A3 = 2 A0 + A1 - 1
(9)
Here, σ0 is the material flow stress, which Newman proposed to be Rp0.2 + Rm (10) σ0 = 2 where Rp0.2 is the monotonic yield stress and Rm is the material ultimate tensile strength. In this paper, σmax is defined as the maximum tangential stress at the notch root for the uncracked specimen, and σ0 is calculated based on equation (10). The effect of three-dimensional constraint is described via a constraint factor α, corresponding to α = 1 under plane stress conditions and α = 3 under plane strain conditions. An approximate equation describing the constraint factor α based on FEA by Newman [20] is written as: 1.5
α = 1.15 + βe- γKn
(11)
where K n K max 0 B , β = 1.25, and γ = 0.85. Here ,B represents specimen thickness, which is considered to be 2.5 mm for the current specimen. 5.2 Aluminum specimen Table 2 shows details of the loading and stress for the three aluminum specimens investigated in this paper, where σmax represents the maximum tangential stress along the notch root in uncracked specimens, and σ'y is the material cyclic yielding stress. The ratio σmax/σ'y which is used to assess the plasticity level for specimens A8 and A7, was 0.378, 0.453, respectively. The maximum tangential stresses in the specimens were all lower than the material cyclic yielding stress and met small scale yielding requirements. The number of cycles was computed on the basis of equation (12), which can be integrated straightforwardly as follows:
N
0
dN =
af
ai
1 da C (ΔKeff )m
(12) The calculation result for specimen A8 is shown in figure 12(a), which shows loading cycles versus crack length c in the notch thickness direction and loading cycles versus crack length a on the outer surface. From the crack path behavior analysis, three fracture modes—tensile, transition, and shear—were determined for this specific crack path. The calculated number of cycles agrees satisfactorily with the experimental data. In the simulation, cracking in the thickness direction arrested at c=2.2mm, corresponding to 182570 cycles. The number of loading cycles expended along this direction in the experiment was 175010. Crack growth occurred on the outer surface, as shown in figure 12(a) by the two dashed lines, which highlight the crack length corresponding to the different fracture modes. Tensile mode occurred from crack length a = 0 mm to a = 16.72 mm (point A). From a = 16.72 mm to a = 26.49
mm (point B), the crack was controlled by transition mode fracture. When the crack was longer than 26.49 mm, shear mode fracture occurred. The slope of loading cycles versus crack length curve can be used to evaluate the crack growth rate. In the case of tensile mode fracture, the slope of the simulated curve increased slightly with crack propagation. Generally, crack growth rates in the simulations were similar to those observed in the experiments. The cycle numbers under tensile mode fracture in the simulation and those observed in the experiment were 387766 and 353000, respectively. When entering transition mode, the crack growth rate increased continually as shown by the simulated curve, whereas the slope of the experimental curve appeared to remain the same as the slope in the tensile mode stage. The numbers of cycles at the end of the transition mode was 418082 in simulation and 445000 in the experiment. Figure 12(b) shows the number of cycles versus crack length curves for specimen A7. Along the thickness direction, the simulated crack growth curve exactly matched the experimental data. Crack length reached c = 2.11 mm after 80541 cycles. In the experiment, 77000 cycles were consumed in the thickness direction, corresponding to c = 2.5 mm. As discussed above, crack extension conforms to tensile mode fracture in the notch thickness direction. When the crack extrapolated to the outer surface of the specimen, the tensile mode controlled crack grew steadily until a = 11.29 mm. The slope of the simulated curve and that of the experimental curve are almost the same, with only a slight difference with increasing crack growth. The number of cycles corresponding to tensile mode fracture was 193134 in the simulation and 175010 in the experiment. The transition mode was shorter than that in specimen A8, from a = 11.27 mm (point A) to a = 15.26 mm (point B).The crack growth rate in the simulation was slightly lower than the real crack propagation rate. Loading cycles increased to 212517 from tensile mode to transition mode based on the calculation, and 195010 cycles were observed during the real crack growth, corresponding to a = 15.5 mm. From both the simulation and the experimental data, the crack growth rate increased when it appeared to be controlled by shear mode fracture. The predicted fatigue life was 234381 cycles at final crack length a = 23.49 mm. The real fatigue life in the experiment was 242721 cycles, at a = 51.0 mm. Crack length under tensile mode fracture occupied only approximately 20% of the total crack length, but more than 70% of loading cycles were consumed under tensile mode fracture. This result is the same as that for specimen A8 and implies that mode I fracture is the dominant fracture mode even in such complex mixed mode loading cases. 5.3 Steel specimen Loading and stress information is shown in table 3, including the maximum tangential stress along the notch root and material cyclic yielding stress for specimens S13 and S7, indicating that these two specimens are tested under relatively high loading levels. The ratio σmax/σ'y was 1.17 for specimen S13 and 1.57 for specimen S7. Both values were above 1. Small scale yielding requirements for
LEFM may not be satisfied, particularly for specimen S7. Figure 13(a) shows the simulated crack growth curve and the experimental data for specimen S13. An obvious distinction between these results and those obtained for the aluminum specimens discussed above is that large deviations exist between the calculated crack growth cycles and the experimentally obtained data along the thickness direction. The experimental curve is sharper than the simulated curve, which means that the crack grew faster in the experiment than in the simulation. From crack initiation to crack growth through the notch thickness, 14000 cycles were expended in the experiment. In contrast, the number of calculated loading cycles was 35015 (corresponding to crack length c = 2.13 mm), which is double the real number of cycles. Plasticity, which was not taken into account in the simulation, may be the main reason for the acceleration of crack growth observed in the notch region. The residual stresses caused by inhomogeneous plastic deformations, which is not considered in this simulation may also contribute to this phenomenon. As the crack extended to the outer surface and propagated continually, the notch effect weakened. The crack was not affected by the notch effect when the crack length was approximately equal to the notch root radius, which was 2 mm in the present specimen. It can be seen from the crack growth curve for crack length a that the calculated curve appears parallel to the experimental curve from a = 2 mm. These parallel curves indicate that the region near the crack front can be treated as linearly elastic when the crack reached a particular length even under relatively high loading cases. From the experimental data, approximately 69% of fatigue life was spent during tensile mode fracture growth. This supports the conclusion that mode I fracture was the dominant fracture mode. A similar phenomenon was observed in specimen S7. Larger deviations were found between the simulated crack growth curve and the experimental crack growth curve in the notch thickness direction. Parallel curves were observed in crack growth along the a direction. The number of calculated crack growth cycles consumed in the c direction was 33288, corresponding to c = 2.11 mm, which is nearly six times more than the 5515 cycles observed in the experiment. Considering that specimen S7 was subjected to a higher loading level than that of specimen S13, the effect of plasticity and residual stresses were expected to be intensified. The acceleration of crack propagation in the notch region was more obvious in this specimen. Because of the plasticity and residual stresses, the simulation overestimated the fatigue life as 73544 cycles, compared with 27400 cycles in the experiment. During the crack growth, including crack propagation in the thickness direction and on the outer surface, mode I stress intensity factors played a dominant role in crack extension because they were approximately 10 times larger than the mode II values throughout crack growth. Tensile mode fracture therefore appropriately describes the entire crack path. No fracture mode transition was detected. 6. Fracture mode transition
When discussing crack growth under mixed mode loading, fracture mode transition must be considered. The transition from shear to tension is published in literature[17], while for the transition from tension to shear, few literatures can be found and the criterion or the theory to describe this behavior is absent. As described in the above analysis and discussion, the employed MTS criterion or the MSS criterion cannot completely reproduce the whole crack growth mechanism for specimen A8 and A7, due to the fracture mode transition from tension to shear. Many factors influence this transition behavior, for example, material properties, the value of tension and torsion loadings, the loading level, the phase angle of loadings and the material properties. An key factor for the current specimen is the geometry. A special geometry that had an increasing thickness was introduced into the current specimen as shown in figure 14. The increasing thickness meant that there was more material that restricted crack growth in a relatively narrow region. The crack paths were forced to become perpendicular to the longitudinal axis to reduce the propagation resistance and make the fracture mode transition from tension to shear. 7. Conclusion In the present paper, 3-dimensional crack growth simulation is implemented by using the LEFM-based algorithm. The simulation results including two out phase angles loading of 45° and 90°. Fracture mode transition from tensile to shear was observed in the crack path behavior analysis. Crack paths belonging to the transition stage cannot be predicted by either the MTS criterion or the MSS criterion. The actual crack path is located somewhere between the predictions of these two criteria. In other specimens, crack growth is only controlled by the MTS criterion, no fracture mode transition is observed. The estimated fatigue lives were computed using Paris law and the effective stress intensity factors. For the small scale yielding specimens (tested under low loading levels), the assessment of loading cycles were in reasonable agreement with the experimental results. For the specimens subjected to higher loading levels, the calculated cycle numbers overestimated the fatigue lives. In these cases, crack growth rates in the notch region were much faster in the experiment than in the simulation. When the crack a on the outer surface of the specimen reached approximately a notch radius (2mm) length, the crack propagation rate in the simulation began to match that in the experiment. ACKNOWLEDGMENT The authors express their gratitude to the Deutsche Forschungsgemeinschaft (DFG) for providing financial support under grant VO729/13-1 and the Fundamental Research Funds for the Central Universities (WUT: 2015IVA022). REFERENCES [1] Abaqus Analysis User’s Manual, Version 6.12-EF. [2] FRANC3D, Fracture Analysis Consultants, Inc., Ithaca, NY, USA; Cornel Fracture
Group, Cornell University, USA. [3] Erdogan, F., Sih, G.C., On the Crack Extension in Plates Under Plane Loading and Transverse Shear. J. Basic Engineering, 85D (1963) 519-527. [4] Sih, G.C., Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fracture, 10 (1974) 305-321. [5] R. J. Nuismer, An energy release rate criterion for mixed mode fracture. Int. J. Fracture, 11 (1975) 245-250. [6] Chambers. A. C., Hyde. T. H., Webster. J. J., Mixed mode fatigue crack growth at 550°C under plane stress conditions in Jethete M152. Engng. Frac. Mech, 39 (1991) 603-619. [7] S.K. Maiti, R.A. Smith, Comparison of the criteria for mixed mode brittle fracture based on the preinstability stress-strain field, Part I: Slit and elliptical cracks under uniaxial tensile loading.Int. J. Fracture, 23 (1983) 281-295. [8] S.K. Maiti, R.A. Smith, Comparison of the criteria for mixed mode brittle fracture based on the preinstability stress-strain field, Part II: Pure shear and uniaxial compressive loading. Int. J. Fracture, 24 (1984) 5-22. [9] Maccagno, T.M., Knott, J.F., The mixed mode I/II fracture behaviour of lightly tempered HY 130 steel at room temperature. Engng. Frac. Mech. 41 (1992) 805-820. [10] Highsmith Jr., S., Crack path determination for non-proportional mixed-mode fatigue. PhD Thesis, Georgia Institute of Technology, USA, (2009). [11] Bold, P.E., Brown, M.W. and Allen, R.J. A review of fatigue crack growth in steels under mixed mode I and II loading. Fatigue Fract. Engng Mater. Struct. 150(1992) 965-977. [12] P. Zerres, M. Vormwald, Review of fatigue crack growth under non-proportional mixed-mode. Int. J. Fatigue, 58 pp. 75-83. 2014. [13] Brüning, J., Untersuchungen zum Rissfortschrittsverhalten unter nichtproportionaler Belastung bei elastisch-plastischem Materialverhalten-Experimente und Theorie. Institut für Stahlbau und Werkstoffmechanik der Technischen Universität Darmstadt, report 85 ISBN 978-3-939195-14-6. (2008) [14] P. Zerres, J.Brüning, M. Vormwald, Fatigue crack growth behavior of fine-grained steel S460N under proportional and non-proportional loading. Eng Fract Mech. 77 (2010) 1822-1834. [15] P. Zerres, J. Brüning, M. Vormwald, Risswachstumsverhalten der Aluminiumlegierung AlMg4.5M unter proportionaler und nicht-proportionaler Schwingbelastung. Mat. Testing, 03 (2011) 109-117. [16] M. Vormwald, T.Seeger, Rißausbreitungsverhalten der Werkstoffe StE460 und AlMg4.5Mn. Report FD-1/1986, Fachgebiet Werkstoffmechanik,TU Darmstadt (1986). [17] D. Haboussa, T. Elguedj, B. Leblé, A. Combescure, Simulation of the shear-tensile mode transition on dynamic crack propagations. Int. J. Fracture, 178 (2012) 195-213. [18] J.C. Newman, Jr., A crack opening stress equation for fatigue crack geowth. Int. J. Fracture, 24 (1984) 131-135. [19] G. Savaidis, M. Dankert, T. Seeger, An analytical procedure for predicting
opening loads of cracks at notchs. Fatigue Fract. Engng Mater. Struct. 18 (1995) 425-442. [20] Newman, J. C., Jr, Crews, J. H., Jr, Biglew, C. A., Dawicke, D.S. Variations of a global constraint factor in cracked bodies under tension and bending loads. ASTM STP 1244 (1995) 21-42. [21] M. Vormwald, Ermüdungslebensdauer von Baustahl unter komplexen Beanspruchungsabläufen am Beispiel des S460. Materials Testing, 53 (2011) pp. 98-108. [22] Hos, Yigiter, Freire, José L F, Vormwald, Michael, Measurements of strain fields around crack tips under proportional and non-proportional mixed-mode fatigue loading. Int. J. Fatigue, 89 (2016) pp. 87-98. [23] Hos, Yigiter, Vormwald, Michael, Experimental study of crack growth under non-proportional loading along with first modeling attempts. Int. J. Fatigue, 92 (2016) pp. 426-433.
Figure captions: Fig.1. Specimen and Notch geometry (all dimensions in mm). Fig. 2. Simulation algorithm procedure. Fig. 3. Uncracked finite element model and crack number. Fig. 4. Crack insertion and remeshed local model. Fig. 5. ABAQUS/FRANC3D working flow. Fig. 6. New crack front fit curve Fig. 7. (a) KI and KII of a node in a full loading cycle (b) KI and KII for the whole crack front under F(tmax) and MT(tmax) Fig. 8. (a)A8 Crack paths in experiment (top) and simulation (bottom) (b) S13 Crack paths in experiment (top) and simulation (bottom). Fig. 9. Crack path 3 in A8. Fig. 10. (a)A7 Crack paths in experiment (top) and simulation (bottom) (b) S7 Crack paths in experiment (top) and simulation (bottom). Fig. 11. Crack path 1 in A7. Fig. 12. (a) Crack growth curve (A8, crack 3) (b) Crack growth curve (A7, crack 1). Fig.13. (a) Crack growth curve (S13, crack 1) (b) Crack growth curve (S7, crack 3). Fig.14. Specimen geometry and crack growth region
Fig. 1.
Specimen and Notch geometry (all dimensions in mm)
Fig.2. Simulation algorithm procedure
reference point
(a) Fig. 3.
Uncracked finite element model and crack number
(b)
Fig. 4. Crack insertion and remeshed local model
Fig. 5.
ABAQUS/FRANC3D working flow
Δanode i
Fig. 6.
New crack front fit curve
(a) (b) Fig. 7. (a) KI and KII of a node in a full loading cycle (b) KI and KII for the whole crack front under F(tmax) and MT(tmax)
z
y
x
(a)
(b)
Fig. 8 (a)A8 Crack paths in experiment (top) and simulation (bottom) (b) S13 Crack paths in experiment (top) and simulation (bottom)
Fig .9. Crack path 3 in A8
(a)
(b)
Fig. 10 (a)A7 Crack paths in experiment (top) and simulation (bottom) (b) S7 Crack paths in experiment (top) and simulation (bottom)
Fig .11. Crack path 1 in A7
(a) Fig. 12.
(b)
(a) Crack growth curve (A8, crack 3) (b) Crack growth curve (A7, crack 1)
(a) Fig. 13.
(b)
(a) Crack growth curve (S13, crack 1) (b) Crack growth curve (S7, crack 3)
crack growth region
Fig .14. Specimen geometry and crack growth region
Table captions: Table 1 Mechanical properties. Table 2 Loading and stress for aluminum specimens. Table 3 Loading and stress information for steel specimens.
Table 1 Mechanical properties
E
Rp0.2
Rm
A5
y '
C
m
[MPa]
[-]
[MPa]
[MPa]
[%]
[MPa]
[-]
[-]
AlMg4.5Mn
68000
0.33
169
340
20.2
341
3.5·10-7
2.9
S460N
208500
0.3
500
643
26.2
410
8.8·10-9
3.15
Material
da/dN in [mm/cycle], ΔKeff in [MPa·m0.5]
Table 2 Loading and stress for aluminum specimens
σmax
σ'y
Fmax
MTmax
Phase angle
(kN)
(Nm)
(°)
A8
6.5
72
45
128.8
341
0.378
A7
8.0
96
90
154.6
341
0.453
Specimen
(MPa)
(MPa)
σmax/σ'y
Table 3 Loading and stress information for steel specimens
Fmax
MTmax
Phase angle
σmax
σ'y
(kN)
(Nm)
(°)
(MPa)
(MPa)
S13
22.5
272
45
479
410
1.17
S7
27.0
408
90
644.6
410
1.57
Specimen
σmax/σ'y
Highlights Crack growth simulation under non-proportional mixed-mode loading is performed. Fracture mode transition from tensile to shear is observed in some specimens. Crack paths of transition stage cannot be predicted by either MTS or MSS criterion.
S0142-1123(17)30195-0 http://dx.doi.org/10.1016/j.ijfatigue.2017.04.014 JIJF 4324
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
21 February 2017 26 April 2017 27 April 2017
Please cite this article as: Yang, Y., Vormwald, M., Fatigue crack growth simulation under cyclic non-proportional mixed mode loading, International Journal of Fatigue (2017), doi: http://dx.doi.org/10.1016/j.ijfatigue.2017.04.014
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Fatigue crack growth simulation under cyclic non-proportional mixed mode loading Ying Yang a,b, c*, Michael Vormwald b a
School of Automotive Engineering, Wuhan University of Technology, Wuhan, Hubei, 430070, China b Materials Mechanics Group, Technische Universität Darmstadt, Franziska-Braun-Str. 3, D-64287 Darmstadt, Germany c Hubei Key Laboratory of Advanced Technology for Automotive Components, Wuhan, Hubei, 430070, China Abstract An algorithm based on linear elastic fracture mechanics for three-dimensional fatigue crack growth simulation under non-proportional mixed-mode loading is proposed in the present paper. The crack growth behaviour in thin-walled, hollow cylinders with a notch under combined non-proportional cyclic tension and torsion loadings are investigated with the finite element program ABAQUS and a 3D fracture analysis software FRANC3D. Different mixed-mode crack path prediction criteria are evaluated to estimate the crack growth direction. Fracture mode transition is observed, some influence factors are discussed. Crack growth cycles calculated based on the effective stress intensity factors satisfactorily match the experimental data for specimens tested under lower loading levels. Keywords Fatigue crack growth, Linear elastic fracture mechanics, Non-proportional loading, Mixed mode
*Corresponding author: Ying Yang Tel: +86 18670810648 E-mail address: [email protected]
1. Introduction When a crack encounters a mixed mode loading, the crack growth path differs from the path under a normal tension loading case, and a deviation will occur. Research in this topic has encountered many problems and challenges in both the theoretical and the experimental fields. In recent years, a series of criteria have been proposed for predictions of the crack propagation direction under mixed mode loading. This paper attempts to investigate and explain fatigue crack growth behavior under non-proportional mixed mode loadings, including the crack growth path and fatigue life assessment. Since Erdogan and Sih [3] first proposed the maximum tangential stress (MTS) criterion for mixed mode fracture prediction in the year of 1963, research in the field of mixed mode crack growth has attracted more and more attention. The minimum strain energy density criterion [4] (S-Criterion, Sih 1974), the maximum energy release rate criterion [5] (Nuismer, 1975), the maximum tangential strain criterion [6] (Chambers, 1991) and so on were suggested subsequently. The various criteria do not provide largely different results for the predicted crack propagation paths under monotonic mixed mode loading. This conclusion is supported by a large quantity of experimental data and theoretical analyses [7] [8] (Maiti, 1983; Maiti, 1984), if the cracks growth is controlled by mode I dominated (tension mode) fracture. However, large distinction between crack propagation direction and the predicted crack kink angle were also observed in some recent experiments. In these cases, the crack growth occurred in the plane where KII values are maximum. As a result the maximum shear stress (MSS) criterion [9] was proposed subsequently (Maccagno, 1992). MSS criterion is available for the predictions of crack growth driven by mode II dominated fracture (shear mode). Although all the above mentioned mixed mode fracture criteria have been deduced from monotonic loading case, they can be implemented straightforward to cyclic proportional mixed mode loading [10] (Highsmith, 2009). According to Bold et al. [11] (1992), this is due to the maximum mechanical parameter ranges (stress or strain) under proportional loading is in direct proportion to the maximum value of these parameters in the crack tip region. Fatigue crack growth under non-proportional mixed mode loading is a more complex situation and there are insufficient hypotheses for application. The very limited researches in this field generally focus on experimental analysis. Seven relevant factors which influence the crack growth behavior under non-proportional loading have been identified in a recent review by Vormwald and Zerres [12] (2012). In the present paper, some of these factors (such as mode-mixity, crack closure and different phase angle) are taken into account by an approach based on linear elastic fracture mechanics.
2. Experiment and material information The current paper focuses on the numerical simulation of 3-dimensional fatigue crack growth under non-proportional mixed mode loading. The corresponding
experimental data were obtained by Brüning [13] (2008). They have been discussed by Zerres et. al [14] [15] (2010, 2011). The aluminium alloy AlMg4.5Mn specimens and steel S460N specimens were tested with the geometry as shown in figure 1. The material properties are shown in the following Table 1[13] [16] [21] (Brüning, 2008; Vormwald,1986; Vormwald,2011). Combined non-proportional cyclic tension and torsion loading with phase angles of 45° and 90° were applied to the specimens with an R ratio of -1. The results of 2 aluminium specimens marked as A7 and A8 and 2 steel specimens marked as S13 and S7 are discussed. 3. Simulation procedure Figure 2 illustrates the procedure of this algorithm. Three main modules can be considered as: (a) determining the crack initiation position; (b) calculating of the maximum equivalent stress intensity factor Keq in one load cycle, which is taken as the crack driving force parameter; (c) crack growth process. Module (b) and (c) are repeated until the crack growth paths can be presented clearly. 3.1 Location of crack The crack initiation will occur in the notch root at the position where the locally uniaxial notch stress is maximum in one loading cycle. A 3-dimensional finite element model was created using ABAQUS for stress analysis. The mesh was refined in the crack growth region as shown in Fig. 3 (a). Along the notch root, there are four potential crack initiation sites numbered as 1, 2, 3 and 4 clockwise, shown in Fig.3 (b). The analysis indicated that the maximum tangential stress in one loading cycle along the notch root on the inside edges is almost 2 times larger than the tangential stress on the outside edge. Therefore a corner crack is considered to be a reasonable shape to initial crack simulation. As can be expected, the initial corner crack will propagate through the thickness of the notch and extend to the outside surface of the specimen in the crack growth process. An incipient corner crack was inserted into the inside edge of the notch root where the tangential stress is maximum by program Fracture Analysis Code 3D (FRANC3D), see figure 4. This program package is designed to simulate crack growth in engineering structures or components with arbitrary crack geometry under different loading and boundary conditions. FRANC3D adaptively remeshes an existing finite element model with the inserted initial crack to generate a cracked model. Furthermore, FRANC3D is also used to calculate stress intensity factor (SIF) for an entire crack front. 3.2 Crack driving force calculation The uncracked finite element model in the current simulation was created in ABAQUS. The typical working flow for ABAQUS/FRANC3D analysis is illustrated in figure 5. In ABAQUS the whole model is divided as the local model and the global model. Boundary conditions and loadings are applied on the global model. The initial cracks are inserted in the local model and cracks only grow in the local model,
so the local model is also called as the crack growth region. The local model is inputted to the FRANC3D to insert the initial cracks in the first step and to calculate SIFs, extend the cracks in the following steps.The cracked local model needs to be merged with the global model, so that the stress analysis can be performed in ABAQUS. Stress analysis results for the cracked model from ABAQUS are used to calculate stress intensity factors in FRANC3D. In the present paper, mode I and mode II SIFs, KI and KII in one cycle under non-proportional tension combined with torsion loading are computed for the sake of obtaining the equivalent SIF, Keq. Erdogan and Sih proposed that the equivalent stress intensity factor according to the maximum tangential stress criterion can be described by:
1 * 3 * 3 * 3 * Keq 3cos cos K sin sin K 4 2 2 4 2 2
(1)
θ* is a function of KI and KII, representing the direction where the tangential stress σθθ in the vicinity of the crack tip is maximum. And also the shear stress τrθ vanishes along this direction on the basis of the MTS criterion. 2 1 K 1 K 2arctan 8 4 K 4 K *
(2)
The stress intensity factors for mode I and mode II are calculated by FRANC3D software for a unit load in tension F=1 and in torsion MT=1. The combined tension and torsion mixed mode stress intensity factors KI and KII are obtained from the superposition according to equation (3):
K I t K I,F F t K I,M M T t K II t K II,F F t K II,M M T t
(3)
The combined SIFs computed from equation (3) are inserted to equation (2) to find the angle θ*, which is put into equation (1). The equivalent stress intensity factor Keq is calculated for a full loading cycle. Among all the equivalent stress intensity factors, the peak value Keq,max is taken as the crack driving force, which means the loads at this instant tmax, F(tmax) and MT(tmax) are used for crack propagation simulation. 3.3 Crack growth process The load corresponding to the Keq,max, F(tmax) and MT(tmax) are applied to the cracked finite element model. Stress analysis is carried out in ABAQUS. The SIF solutions for the cracks are again obtained from FRANC3D based on the stress analysis. Although the solutions for KI, KII and Keq are already at hand for t= tmax, the
re-evaluation of this loading case combination can now be used to let the FRANC3D software create the crack propagation increment. In FRANC3D, a 3-dimensional crack front is a space curve comprising a series of extrapolated nodes. For each node, the corresponding increment Δanode i is a value in a given proportion to a user defined crack growth increment Δauser. This proportion is related to the stress intensity factors along the crack front. A new node is extended in front of the old node with a distance of Δanode i as shown in figure 6 calculated based on equation (4).
Δanode i
ΔKnode i = Δauser ΔKeq,mean
n
(4)
The ABAQUS/FRANC3D interface was implemented to perform the crack growth analysis. The finite element modeling and the stress analysis are performed in ABAQUS, FRANC3D is used to calculate crack growth parameters and updates the crack geometry and mesh. This process is continued until the crack has grown to a certain length. The KI and KII of a node in the crack front and the KI and KII for the whole crack front under F(tmax) and MT(tmax) for the initial crack of specimen A8 are shown in figure 7. For the intial crack of specimen A8, the maximum mode I stress intensity factor component is more than 10 times larger than the maximum mode II values. This indicates that the mode II fracture will not affect crack growth behavior in the full cycle, mode I fracture is determined. For this reason, the MTS criterion is used as the primary rule for estimating the crack propagation direction in the crack growth simulation. The MSS criterion is performed as an alternative formula applicable to mode II controlled crack growth. In some cases, the simulated crack growth path deviated from the experimental crack path no matter what criterion was employed. In order to investigate the crack growth path behavior under these complicated loading cases, a modification is designed to keep the deviation between experimental results and simulation in an acceptable extent. The modification is implemented manually to adjust the position of the newly generated crack front in order to make the simulated crack increment direction coincident with the experimental results. As a result, although the simulated crack growth path is not perfectly identical with the experimental data, a close match is enforced. 4. Crack growth paths 4.1 Non-proportional loading with phase angle of 45° An aluminum specimen labeled A8 (AlMg4.5Mn; Fmax = 6.5 kN; MTmax = 72 Nm) was tested under non-proportional mixed mode loading with a phase angle of 45°. Figure 8(a) shows the crack propagation paths on the outer surface of the specimen in the experiment and in the simulation. As mentioned before, four potential crack initiation sites at the notch root were
numbered 1, 2, 3, and 4 clockwise (see figure 3). The corresponding cracks are called crack 1, 2, 3, and 4, respectively. Two crack paths were observed in specimen A8. Crack 1 is shown on the left of figure 8(a), and crack 3 is on the right. Cracks initiated at sites 1 and 3 because the maximum tangential stresses at these two sites were larger than those at sites 2 and 4 in one loading cycle. In the simulation, the initiation angle was approximately 38.4° (-38.4°). As can be seen in the figure, the crack propagation trajectories in the simulation were similar to those in the experiment. Cracks propagated along an approximate angle of 40° to the longitudinal axis of the specimen initially and then deflected gradually until the paths transformed into planes, which were perpendicular to the longitudinal axis. Because the two crack growth trajectories showed antisymmetry in both the experiment and the simulation, the behaviors of these two cracks should coincide. Taking crack 3 to analyze the crack path behavior, the simulated crack 3 can be seen in figure 9. Three stages of the crack path are shown in this figure: The crack path from notch root to point A was predicted by the MTS criterion in FRANC3D; that from point B to the end was predicted by the MSS criterion in FRANC3D; and the path AB was fitted from the experimental result. To provide a clear comparison between the paths predicted by the different criteria, two estimated results are also shown in figure 9. The path indicated by squares is the estimated result calculated based on the MTS criterion, and the path indicated by triangles is that based on the MSS criterion. Until point A, the crack propagation trajectory was satisfactorily estimated by the MTS criterion. When the MTS criterion was applied continuously, the estimated solution showed a downward crack path propagating in nearly the original direction with slight deflection. However, from point A, the real crack path deviated from the prediction based on the MTS criterion. This implies that the crack propagation direction was no longer normal to the maximum tangential stress in the vicinity of the crack tip. Another criterion was therefore used to attempt to calculate the crack kink angle from this point. The MSS criterion was applied here as an alternative rule to predict the crack path. The crack path starting from point A clearly did not follow the route estimated by either the MTS criterion or the MSS criterion based on the estimated solutions. The path according to the MSS criterion showed an upward trend from the original crack growth direction. A path somewhere between the predictions of the MTS and MSS criteria that gradually deviated from the original direction appears to be more reasonable. At present, no criteria are available that can be used to calculate crack path AB. As a result, path AB in FRANC3D is a fitting curve based on the experimental data. From point B, the crack path flattened (perpendicular to the longitudinal axis of the specimen), and the MSS criterion provided a successful prediction. The crack path calculated in FRANC3D matched the real crack path well. In contrast, large divergence was observed when the MTS criterion was used to predict the crack path, as shown in figure 9, from the estimated solution. Generally, the MTS criterion predicted that the crack path maintained a nearly straight line with a constant angle to the longitudinal axis, with slight deflection. The flat crack path was satisfactorily described by the MSS criterion. As we already
know, based on the LEFM analysis, the MTS criterion is valid for tensile mode fractures, and the MSS criterion is used for shear mode problems. Both of these criteria are supported by a large quantity of corresponding experiments. Based on the crack path behavior analysis, two distinct fracture modes and fracture mode transition were observed in the experiment. A steel specimen labeled S13 (S460N; Fmax = 22.5 kN; MTmax = 272 Nm) was subjected to cyclic tension combined with torsion loading with an out-of-phase angle of 45°. The crack paths in the experiment and those in the simulation are shown in figure 8(b). Crack 1 is on the left of this figure, and crack 3 is on the right. The crack propagation trajectories were similar to the paths in specimen A8. Cracks initiated at sites 1 and 3. The angle ϕ calculated based on finite element stress analysis was approximately 40.6° (-40.6°). Initially, both cracks propagated along an angle of approximately 40° (-40°) to the longitudinal axis. The final paths flattened and were perpendicular to the longitudinal axis of the specimen. Fracture mode transition from tensile to shear was also observed in this specimen. 4.2 Non-proportional loading with phase angle of 90° The other non-proportional loading tested in the experiment was with an out-of-phase loading angle of 90°. Figure 10(a) shows photographs of crack paths in an aluminum specimen labeled A7 (AlMg4.5Mn; Fmax = 8.0 kN; MTmax = 96 Nm) and the simulated crack paths. Two cracks were observed: crack 1 (on the left in the figure) and crack 3 (on the right of the figure). From the photographs, it was found that both crack paths were very curvilinear, and crack branching was observed in the experiment. When this test was performed, the simulation algorithm did not cover crack branching, and therefore, the simulation stopped before any crack branching could occur. According to the stress analysis, the crack initiation angle ϕ was 51.9°. Taking crack 1 to analyze the crack path behavior, the path first propagated upward following a smooth curve, and as the crack grew, the extension direction changed to downward, which was opposite to the original orientation. The crack paths in this specimen were different from those in specimens under loading with a phase angle of 45°. No flat crack path was observed in the experiment. The result of analysis of crack path behavior is shown in figure 11. From the notch root to point A, the smooth crack path can be described well by the MTS criterion. The corresponding fracture was tensile mode fracture. Deviations between the simulated path and the experimental path became evident as the crack extended. Starting from point A, the MSS criterion was used as the alternative rule to predict crack growth direction. Estimation shows that both the prediction from the MTS criterion and that from the MSS criterion did not provide acceptable results, because the real crack path was somewhere between these two estimations. When the crack length reached point B, the downward crack path was fully controlled by the MSS criterion. From this moment, the crack was controlled under shear mode fracture. Path AB was considered to signify transition mode and was fitted from the
experimental data. Steel specimen S7 was tested under tension combined with torsion loading with Fmax = 27 kN and MTmax = 408 Nm. Four cracks were observed in the experiment. In specimen S7, four crack paths were observed to be nearly symmetrical in both the experiment and the simulation, as shown in figure 10(b). According to the FEA, the maximum tangential stresses along the notch root at sites 1to 4 were nearly equal. The calculated crack initiation angle was 53.1° (-53.1°). Because the four cracks initiated at nearly the same loading cycles in the experiment, initial cracks of the same size were inserted into the corresponding positions in FRANC3D. The entire simulated crack path was calculated from FRANC3D based on the MTS criterion, which showed that crack propagation complied with tension controlled fracture, and no fracture mode transition took place. 5. Crack growth cycles Fatigue life includes the loading cycles expended in crack initiation and those expended during crack propagation. In the present paper, the number of cycles expended from the initial crack to that of the final crack is calculated based on Paris law, the number of loading cycles consumed in crack initiation is obtained from experimental data. 5.1 Crack closure effect The maximum equivalent stress intensity factor Keq,max in one loading cycle is used to calculate the instant forces corresponding to crack propagation. Under cyclic loading, the stress intensity factor range ΔK is associated with the crack deflection angle as well as fatigue life. When loading ratio R = -1, crack closure must affect fatigue life significantly. Usually, ΔK is substituted for the effective stress intensity factor range ΔKeff to take account of the loading ratio effect. Here, ΔKeff is defined as Kmax − Kop, where the subscript “op” denotes a crack opening stress intensity factor. Newman’s crack opening stress equation [18] is one of the most widely used methods for estimating opening stress and is given by: σop σmax σop σmax
= A0 + A1 R + A2 R2 + A3 R3
for R > 0
(5) for - 2 R < 0
= A0 + A1 R
Where
πσ A0 = 0.825 - 0.34α + 0.05α2 cos max 2σ0 A1 = (0.415 - 0.071α)
A2 = 1- A0 - A1 - A3
σmax σ0
1
α
(6)
(7)
(8)
A3 = 2 A0 + A1 - 1
(9)
Here, σ0 is the material flow stress, which Newman proposed to be Rp0.2 + Rm (10) σ0 = 2 where Rp0.2 is the monotonic yield stress and Rm is the material ultimate tensile strength. In this paper, σmax is defined as the maximum tangential stress at the notch root for the uncracked specimen, and σ0 is calculated based on equation (10). The effect of three-dimensional constraint is described via a constraint factor α, corresponding to α = 1 under plane stress conditions and α = 3 under plane strain conditions. An approximate equation describing the constraint factor α based on FEA by Newman [20] is written as: 1.5
α = 1.15 + βe- γKn
(11)
where K n K max 0 B , β = 1.25, and γ = 0.85. Here ,B represents specimen thickness, which is considered to be 2.5 mm for the current specimen. 5.2 Aluminum specimen Table 2 shows details of the loading and stress for the three aluminum specimens investigated in this paper, where σmax represents the maximum tangential stress along the notch root in uncracked specimens, and σ'y is the material cyclic yielding stress. The ratio σmax/σ'y which is used to assess the plasticity level for specimens A8 and A7, was 0.378, 0.453, respectively. The maximum tangential stresses in the specimens were all lower than the material cyclic yielding stress and met small scale yielding requirements. The number of cycles was computed on the basis of equation (12), which can be integrated straightforwardly as follows:
N
0
dN =
af
ai
1 da C (ΔKeff )m
(12) The calculation result for specimen A8 is shown in figure 12(a), which shows loading cycles versus crack length c in the notch thickness direction and loading cycles versus crack length a on the outer surface. From the crack path behavior analysis, three fracture modes—tensile, transition, and shear—were determined for this specific crack path. The calculated number of cycles agrees satisfactorily with the experimental data. In the simulation, cracking in the thickness direction arrested at c=2.2mm, corresponding to 182570 cycles. The number of loading cycles expended along this direction in the experiment was 175010. Crack growth occurred on the outer surface, as shown in figure 12(a) by the two dashed lines, which highlight the crack length corresponding to the different fracture modes. Tensile mode occurred from crack length a = 0 mm to a = 16.72 mm (point A). From a = 16.72 mm to a = 26.49
mm (point B), the crack was controlled by transition mode fracture. When the crack was longer than 26.49 mm, shear mode fracture occurred. The slope of loading cycles versus crack length curve can be used to evaluate the crack growth rate. In the case of tensile mode fracture, the slope of the simulated curve increased slightly with crack propagation. Generally, crack growth rates in the simulations were similar to those observed in the experiments. The cycle numbers under tensile mode fracture in the simulation and those observed in the experiment were 387766 and 353000, respectively. When entering transition mode, the crack growth rate increased continually as shown by the simulated curve, whereas the slope of the experimental curve appeared to remain the same as the slope in the tensile mode stage. The numbers of cycles at the end of the transition mode was 418082 in simulation and 445000 in the experiment. Figure 12(b) shows the number of cycles versus crack length curves for specimen A7. Along the thickness direction, the simulated crack growth curve exactly matched the experimental data. Crack length reached c = 2.11 mm after 80541 cycles. In the experiment, 77000 cycles were consumed in the thickness direction, corresponding to c = 2.5 mm. As discussed above, crack extension conforms to tensile mode fracture in the notch thickness direction. When the crack extrapolated to the outer surface of the specimen, the tensile mode controlled crack grew steadily until a = 11.29 mm. The slope of the simulated curve and that of the experimental curve are almost the same, with only a slight difference with increasing crack growth. The number of cycles corresponding to tensile mode fracture was 193134 in the simulation and 175010 in the experiment. The transition mode was shorter than that in specimen A8, from a = 11.27 mm (point A) to a = 15.26 mm (point B).The crack growth rate in the simulation was slightly lower than the real crack propagation rate. Loading cycles increased to 212517 from tensile mode to transition mode based on the calculation, and 195010 cycles were observed during the real crack growth, corresponding to a = 15.5 mm. From both the simulation and the experimental data, the crack growth rate increased when it appeared to be controlled by shear mode fracture. The predicted fatigue life was 234381 cycles at final crack length a = 23.49 mm. The real fatigue life in the experiment was 242721 cycles, at a = 51.0 mm. Crack length under tensile mode fracture occupied only approximately 20% of the total crack length, but more than 70% of loading cycles were consumed under tensile mode fracture. This result is the same as that for specimen A8 and implies that mode I fracture is the dominant fracture mode even in such complex mixed mode loading cases. 5.3 Steel specimen Loading and stress information is shown in table 3, including the maximum tangential stress along the notch root and material cyclic yielding stress for specimens S13 and S7, indicating that these two specimens are tested under relatively high loading levels. The ratio σmax/σ'y was 1.17 for specimen S13 and 1.57 for specimen S7. Both values were above 1. Small scale yielding requirements for
LEFM may not be satisfied, particularly for specimen S7. Figure 13(a) shows the simulated crack growth curve and the experimental data for specimen S13. An obvious distinction between these results and those obtained for the aluminum specimens discussed above is that large deviations exist between the calculated crack growth cycles and the experimentally obtained data along the thickness direction. The experimental curve is sharper than the simulated curve, which means that the crack grew faster in the experiment than in the simulation. From crack initiation to crack growth through the notch thickness, 14000 cycles were expended in the experiment. In contrast, the number of calculated loading cycles was 35015 (corresponding to crack length c = 2.13 mm), which is double the real number of cycles. Plasticity, which was not taken into account in the simulation, may be the main reason for the acceleration of crack growth observed in the notch region. The residual stresses caused by inhomogeneous plastic deformations, which is not considered in this simulation may also contribute to this phenomenon. As the crack extended to the outer surface and propagated continually, the notch effect weakened. The crack was not affected by the notch effect when the crack length was approximately equal to the notch root radius, which was 2 mm in the present specimen. It can be seen from the crack growth curve for crack length a that the calculated curve appears parallel to the experimental curve from a = 2 mm. These parallel curves indicate that the region near the crack front can be treated as linearly elastic when the crack reached a particular length even under relatively high loading cases. From the experimental data, approximately 69% of fatigue life was spent during tensile mode fracture growth. This supports the conclusion that mode I fracture was the dominant fracture mode. A similar phenomenon was observed in specimen S7. Larger deviations were found between the simulated crack growth curve and the experimental crack growth curve in the notch thickness direction. Parallel curves were observed in crack growth along the a direction. The number of calculated crack growth cycles consumed in the c direction was 33288, corresponding to c = 2.11 mm, which is nearly six times more than the 5515 cycles observed in the experiment. Considering that specimen S7 was subjected to a higher loading level than that of specimen S13, the effect of plasticity and residual stresses were expected to be intensified. The acceleration of crack propagation in the notch region was more obvious in this specimen. Because of the plasticity and residual stresses, the simulation overestimated the fatigue life as 73544 cycles, compared with 27400 cycles in the experiment. During the crack growth, including crack propagation in the thickness direction and on the outer surface, mode I stress intensity factors played a dominant role in crack extension because they were approximately 10 times larger than the mode II values throughout crack growth. Tensile mode fracture therefore appropriately describes the entire crack path. No fracture mode transition was detected. 6. Fracture mode transition
When discussing crack growth under mixed mode loading, fracture mode transition must be considered. The transition from shear to tension is published in literature[17], while for the transition from tension to shear, few literatures can be found and the criterion or the theory to describe this behavior is absent. As described in the above analysis and discussion, the employed MTS criterion or the MSS criterion cannot completely reproduce the whole crack growth mechanism for specimen A8 and A7, due to the fracture mode transition from tension to shear. Many factors influence this transition behavior, for example, material properties, the value of tension and torsion loadings, the loading level, the phase angle of loadings and the material properties. An key factor for the current specimen is the geometry. A special geometry that had an increasing thickness was introduced into the current specimen as shown in figure 14. The increasing thickness meant that there was more material that restricted crack growth in a relatively narrow region. The crack paths were forced to become perpendicular to the longitudinal axis to reduce the propagation resistance and make the fracture mode transition from tension to shear. 7. Conclusion In the present paper, 3-dimensional crack growth simulation is implemented by using the LEFM-based algorithm. The simulation results including two out phase angles loading of 45° and 90°. Fracture mode transition from tensile to shear was observed in the crack path behavior analysis. Crack paths belonging to the transition stage cannot be predicted by either the MTS criterion or the MSS criterion. The actual crack path is located somewhere between the predictions of these two criteria. In other specimens, crack growth is only controlled by the MTS criterion, no fracture mode transition is observed. The estimated fatigue lives were computed using Paris law and the effective stress intensity factors. For the small scale yielding specimens (tested under low loading levels), the assessment of loading cycles were in reasonable agreement with the experimental results. For the specimens subjected to higher loading levels, the calculated cycle numbers overestimated the fatigue lives. In these cases, crack growth rates in the notch region were much faster in the experiment than in the simulation. When the crack a on the outer surface of the specimen reached approximately a notch radius (2mm) length, the crack propagation rate in the simulation began to match that in the experiment. ACKNOWLEDGMENT The authors express their gratitude to the Deutsche Forschungsgemeinschaft (DFG) for providing financial support under grant VO729/13-1 and the Fundamental Research Funds for the Central Universities (WUT: 2015IVA022). REFERENCES [1] Abaqus Analysis User’s Manual, Version 6.12-EF. [2] FRANC3D, Fracture Analysis Consultants, Inc., Ithaca, NY, USA; Cornel Fracture
Group, Cornell University, USA. [3] Erdogan, F., Sih, G.C., On the Crack Extension in Plates Under Plane Loading and Transverse Shear. J. Basic Engineering, 85D (1963) 519-527. [4] Sih, G.C., Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fracture, 10 (1974) 305-321. [5] R. J. Nuismer, An energy release rate criterion for mixed mode fracture. Int. J. Fracture, 11 (1975) 245-250. [6] Chambers. A. C., Hyde. T. H., Webster. J. J., Mixed mode fatigue crack growth at 550°C under plane stress conditions in Jethete M152. Engng. Frac. Mech, 39 (1991) 603-619. [7] S.K. Maiti, R.A. Smith, Comparison of the criteria for mixed mode brittle fracture based on the preinstability stress-strain field, Part I: Slit and elliptical cracks under uniaxial tensile loading.Int. J. Fracture, 23 (1983) 281-295. [8] S.K. Maiti, R.A. Smith, Comparison of the criteria for mixed mode brittle fracture based on the preinstability stress-strain field, Part II: Pure shear and uniaxial compressive loading. Int. J. Fracture, 24 (1984) 5-22. [9] Maccagno, T.M., Knott, J.F., The mixed mode I/II fracture behaviour of lightly tempered HY 130 steel at room temperature. Engng. Frac. Mech. 41 (1992) 805-820. [10] Highsmith Jr., S., Crack path determination for non-proportional mixed-mode fatigue. PhD Thesis, Georgia Institute of Technology, USA, (2009). [11] Bold, P.E., Brown, M.W. and Allen, R.J. A review of fatigue crack growth in steels under mixed mode I and II loading. Fatigue Fract. Engng Mater. Struct. 150(1992) 965-977. [12] P. Zerres, M. Vormwald, Review of fatigue crack growth under non-proportional mixed-mode. Int. J. Fatigue, 58 pp. 75-83. 2014. [13] Brüning, J., Untersuchungen zum Rissfortschrittsverhalten unter nichtproportionaler Belastung bei elastisch-plastischem Materialverhalten-Experimente und Theorie. Institut für Stahlbau und Werkstoffmechanik der Technischen Universität Darmstadt, report 85 ISBN 978-3-939195-14-6. (2008) [14] P. Zerres, J.Brüning, M. Vormwald, Fatigue crack growth behavior of fine-grained steel S460N under proportional and non-proportional loading. Eng Fract Mech. 77 (2010) 1822-1834. [15] P. Zerres, J. Brüning, M. Vormwald, Risswachstumsverhalten der Aluminiumlegierung AlMg4.5M unter proportionaler und nicht-proportionaler Schwingbelastung. Mat. Testing, 03 (2011) 109-117. [16] M. Vormwald, T.Seeger, Rißausbreitungsverhalten der Werkstoffe StE460 und AlMg4.5Mn. Report FD-1/1986, Fachgebiet Werkstoffmechanik,TU Darmstadt (1986). [17] D. Haboussa, T. Elguedj, B. Leblé, A. Combescure, Simulation of the shear-tensile mode transition on dynamic crack propagations. Int. J. Fracture, 178 (2012) 195-213. [18] J.C. Newman, Jr., A crack opening stress equation for fatigue crack geowth. Int. J. Fracture, 24 (1984) 131-135. [19] G. Savaidis, M. Dankert, T. Seeger, An analytical procedure for predicting
opening loads of cracks at notchs. Fatigue Fract. Engng Mater. Struct. 18 (1995) 425-442. [20] Newman, J. C., Jr, Crews, J. H., Jr, Biglew, C. A., Dawicke, D.S. Variations of a global constraint factor in cracked bodies under tension and bending loads. ASTM STP 1244 (1995) 21-42. [21] M. Vormwald, Ermüdungslebensdauer von Baustahl unter komplexen Beanspruchungsabläufen am Beispiel des S460. Materials Testing, 53 (2011) pp. 98-108. [22] Hos, Yigiter, Freire, José L F, Vormwald, Michael, Measurements of strain fields around crack tips under proportional and non-proportional mixed-mode fatigue loading. Int. J. Fatigue, 89 (2016) pp. 87-98. [23] Hos, Yigiter, Vormwald, Michael, Experimental study of crack growth under non-proportional loading along with first modeling attempts. Int. J. Fatigue, 92 (2016) pp. 426-433.
Figure captions: Fig.1. Specimen and Notch geometry (all dimensions in mm). Fig. 2. Simulation algorithm procedure. Fig. 3. Uncracked finite element model and crack number. Fig. 4. Crack insertion and remeshed local model. Fig. 5. ABAQUS/FRANC3D working flow. Fig. 6. New crack front fit curve Fig. 7. (a) KI and KII of a node in a full loading cycle (b) KI and KII for the whole crack front under F(tmax) and MT(tmax) Fig. 8. (a)A8 Crack paths in experiment (top) and simulation (bottom) (b) S13 Crack paths in experiment (top) and simulation (bottom). Fig. 9. Crack path 3 in A8. Fig. 10. (a)A7 Crack paths in experiment (top) and simulation (bottom) (b) S7 Crack paths in experiment (top) and simulation (bottom). Fig. 11. Crack path 1 in A7. Fig. 12. (a) Crack growth curve (A8, crack 3) (b) Crack growth curve (A7, crack 1). Fig.13. (a) Crack growth curve (S13, crack 1) (b) Crack growth curve (S7, crack 3). Fig.14. Specimen geometry and crack growth region
Fig. 1.
Specimen and Notch geometry (all dimensions in mm)
Fig.2. Simulation algorithm procedure
reference point
(a) Fig. 3.
Uncracked finite element model and crack number
(b)
Fig. 4. Crack insertion and remeshed local model
Fig. 5.
ABAQUS/FRANC3D working flow
Δanode i
Fig. 6.
New crack front fit curve
(a) (b) Fig. 7. (a) KI and KII of a node in a full loading cycle (b) KI and KII for the whole crack front under F(tmax) and MT(tmax)
z
y
x
(a)
(b)
Fig. 8 (a)A8 Crack paths in experiment (top) and simulation (bottom) (b) S13 Crack paths in experiment (top) and simulation (bottom)
Fig .9. Crack path 3 in A8
(a)
(b)
Fig. 10 (a)A7 Crack paths in experiment (top) and simulation (bottom) (b) S7 Crack paths in experiment (top) and simulation (bottom)
Fig .11. Crack path 1 in A7
(a) Fig. 12.
(b)
(a) Crack growth curve (A8, crack 3) (b) Crack growth curve (A7, crack 1)
(a) Fig. 13.
(b)
(a) Crack growth curve (S13, crack 1) (b) Crack growth curve (S7, crack 3)
crack growth region
Fig .14. Specimen geometry and crack growth region
Table captions: Table 1 Mechanical properties. Table 2 Loading and stress for aluminum specimens. Table 3 Loading and stress information for steel specimens.
Table 1 Mechanical properties
E
Rp0.2
Rm
A5
y '
C
m
[MPa]
[-]
[MPa]
[MPa]
[%]
[MPa]
[-]
[-]
AlMg4.5Mn
68000
0.33
169
340
20.2
341
3.5·10-7
2.9
S460N
208500
0.3
500
643
26.2
410
8.8·10-9
3.15
Material
da/dN in [mm/cycle], ΔKeff in [MPa·m0.5]
Table 2 Loading and stress for aluminum specimens
σmax
σ'y
Fmax
MTmax
Phase angle
(kN)
(Nm)
(°)
A8
6.5
72
45
128.8
341
0.378
A7
8.0
96
90
154.6
341
0.453
Specimen
(MPa)
(MPa)
σmax/σ'y
Table 3 Loading and stress information for steel specimens
Fmax
MTmax
Phase angle
σmax
σ'y
(kN)
(Nm)
(°)
(MPa)
(MPa)
S13
22.5
272
45
479
410
1.17
S7
27.0
408
90
644.6
410
1.57
Specimen
σmax/σ'y
Highlights Crack growth simulation under non-proportional mixed-mode loading is performed. Fracture mode transition from tensile to shear is observed in some specimens. Crack paths of transition stage cannot be predicted by either MTS or MSS criterion.